Why Incidence and Prevalence Are Different
A disability insurer may ask two different questions about a group of people. How many people entered disability during a period? This is an incidence question. How many people are disabled at a particular date? This is a prevalence question. The answers need not be equal: someone disabled before the period can still be disabled at the end, and someone who was active at the start can become disabled during it.
This distinction matters when interpreting experience studies and state probabilities. A disability prevalence is a probability of occupying the disabled state at a specified time. An incidence calculation measures transitions into disability over a period, with its denominator stated explicitly. The methods below use a discrete-time state model, with state probabilities recorded at successive observation dates. The tutorial is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
State Probabilities and the One-Period Calculation
Use a row vector for the probabilities of occupying the states at an observation date, and let the rows of a transition matrix represent the state at the start of a period while its columns represent the state at the end. For a two-state model with no exit from the population, write \(p_t=(p_A(t),p_D(t))\) and \(P=\begin{pmatrix}p_{AA}&p_{AD}\\p_{DA}&p_{DD}\end{pmatrix}\). Each row of \(P\) sums to one. The state probabilities at the next observation date are \(p_{t+1}=p_tP\).
Here, \(p_{AD}\) is the probability that someone active at the start is disabled at the end of the period. Thus, the proportion of the original cohort making this transition is \(p_A(t)p_{AD}\). By contrast, the disabled-state probability at the end is \(p_D(t+1)\): it includes people who entered disability from active and people who were already disabled and remain so.
These expressions assume the stated transition probabilities describe the movement between the observation dates and the cohort is represented by the given states. If the model includes a state such as death or withdrawal, the transition matrix must include it, and prevalence per original cohort still uses the original cohort as its denominator. Prevalence among people remaining in the population is a different ratio.
Worked Example: New Entries and End-of-Year Prevalence
A hypothetical disability plan follows a cohort that is entirely active on 1 January. Over each year, an active member is active at year-end with probability \(0.88\) and disabled with probability \(0.12\). A disabled member is active at year-end with probability \(0.30\) and remains disabled with probability \(0.70\). Find the first-year incidence and prevalence, then the second-year incidence and prevalence, each as a proportion of the original cohort.
Method and assumptions. Use the same annual transition matrix in each year; probabilities describe the state at consecutive year-ends. The row vector is ordered \(A,D\), so \(P=\begin{pmatrix}0.88&0.12\\0.30&0.70\end{pmatrix}\). There is no exit state.
At the start, \(p_0=(1,0)\). After one year, \(p_1=p_0P=(0.88,0.12)\). The first-year proportion making an \(A\)-to-\(D\) transition is \(1(0.12)=0.12\). The year-end prevalence is also \(0.12\), because no one was disabled at the start.
For year two, \(p_A(1)=0.88\) and \(p_D(1)=0.12\). The proportion of the original cohort making an \(A\)-to-\(D\) transition in year two is \(0.88(0.12)=0.1056\). The proportion disabled at the end of year two is \(0.88(0.12)+0.12(0.70)=0.1056+0.0840=0.1896\).
Check. The year-two active probability is \(0.88(0.88)+0.12(0.30)=0.7744+0.0360=0.8104\). Therefore \(0.8104+0.1896=1\). The calculations distinguish new transitions, \(0.1056\), from all people disabled at year-end, \(0.1896\). The common mistake avoided is counting everyone still disabled as a new entrant.
Deriving the Prevalence Update
The update formula follows by conditioning on the state at the start of the period. A person can be disabled at the end either by starting active and moving to disabled, or by starting disabled and remaining disabled. These are disjoint routes to the end-of-period disabled state, so their probabilities add.
In the two-state model, a person starts either active with probability \(p_A(t)\), or disabled with probability \(p_D(t)\).
The active-to-disabled route has probability \(p_A(t)p_{AD}\). The disabled-to-disabled route has probability \(p_D(t)p_{DD}\).
Thus \(p_D(t+1)=p_A(t)p_{AD}+p_D(t)p_{DD}\). The first term is new incidence from the active state; the second is continuing disability.
This derivation also identifies why prevalence cannot generally be substituted for incidence. Prevalence is a stock at a date, while incidence is a flow over a period. Recovery affects how much of the existing disabled group remains in the stock; it does not turn those continuing disabled members into new transitions.
Worked Example: Initial Disability and Prevalence Among Survivors
A hypothetical cohort begins with state probabilities \((0.50,0.40,0.10)\) in states \(A,D,X\), where \(X\) is an absorbing exit state. Its one-year transition matrix is \(P=\begin{pmatrix}0.80&0.15&0.05\\0.30&0.60&0.10\\0&0&1\end{pmatrix}\). Find the year-end prevalence per original cohort, new active-to-disabled transitions per original cohort, and disability prevalence among those in \(A\) or \(D\) at year-end.
Method and assumptions. Multiply the starting row vector by the matrix. The rows are starting states and columns are ending states, both ordered \(A,D,X\). State \(X\) is absorbing. The original cohort is the denominator for the first two requested proportions.
The year-end active probability is \(0.50(0.80)+0.40(0.30)+0.10(0)=0.4000+0.1200=0.5200\). The disabled probability is \(0.50(0.15)+0.40(0.60)+0.10(0)=0.0750+0.2400=0.3150\). The exit probability is \(0.50(0.05)+0.40(0.10)+0.10(1)=0.0250+0.0400+0.1000=0.1650\).
New active-to-disabled transitions per original cohort are \(0.50(0.15)=0.0750\). This counts only people who began the year active and ended it disabled under the transition model; the \(0.2400\) contribution from starting disabled and remaining disabled is not new incidence.
The proportion still in \(A\) or \(D\) at year-end is \(0.5200+0.3150=0.8350\). Therefore prevalence among those remaining in these two states is \(0.3150/0.8350=0.3772455\ldots\), or \(0.3772\) to four decimal places. The check is \(0.5200+0.3150+0.1650=1.0000\). The common mistake avoided is dividing by the original cohort when the requested denominator is only those remaining in \(A\) or \(D\).
Conditional Incidence and Changing Transition Probabilities
Incidence can be reported with different denominators. If \(p_A(t)>0\), the transition probability \(p_{AD}\) is incidence conditional on being active at the start of the period. The corresponding proportion of the original cohort is \(p_A(t)p_{AD}\). These quantities answer different questions: one describes risk for the active group; the other describes the cohort contribution.
If transition probabilities vary by period, apply each period's matrix in sequence. For two periods, \(p_2=p_0P_1P_2\). Incidence during the second period is based on the active probability at the start of that period and the second period's active-to-disabled transition probability. It is not calculated from the initial state probabilities unless the period begins at time zero.
Worked Example: A Period-Specific Transition Matrix
A hypothetical cohort is entirely active at the beginning of year one. In year one the transition probabilities are \(A\to A=0.90\), \(A\to D=0.10\), \(D\to A=0.40\), and \(D\to D=0.60\). In year two they are \(A\to A=0.80\), \(A\to D=0.20\), \(D\to A=0.30\), and \(D\to D=0.70\). Find year-two incidence per original cohort, conditional year-two incidence among those active at its start, and prevalence at the end of year two.
Method and assumptions. Use row vectors ordered \(A,D\), and use the matrix for each year only in its corresponding period. At the start \(p_0=(1,0)\). After year one, \(p_1=(1,0)\begin{pmatrix}0.90&0.10\\0.40&0.60\end{pmatrix}=(0.90,0.10)\).
The year-two conditional incidence among those active at its start is the year-two transition probability \(0.20\). Since \(0.90\) of the original cohort is active at the start of year two, year-two incidence per original cohort is \(0.90(0.20)=0.18\).
For year-end prevalence, include both routes to \(D\): \(p_D(2)=0.90(0.20)+0.10(0.70)=0.18+0.07=0.25\). For a check, the active probability is \(0.90(0.80)+0.10(0.30)=0.72+0.03=0.75\), and the two probabilities sum to \(1.00\).
The common mistake avoided is using year-one incidence \(0.10\) as the year-two incidence. Year two has a different transition probability and a different active population at its start.
Interpretation and Common Exam Traps
The same state model can produce several valid numerical measures, so label the date, period, and denominator whenever reporting a result. Prevalence per original cohort is simply the disabled-state probability. Prevalence among survivors or continuing members divides the disabled probability by the probability of being in the states included in that denominator. Conditional incidence among active members is \(p_{AD}\), while incidence per original cohort is \(p_A(t)p_{AD}\).
A frequent error is to report \(p_D(t+1)\) as the number of new disability cases. For example, if \(p_A(t)=0.50\), \(p_D(t)=0.40\), \(p_{AD}=0.15\), and \(p_{DD}=0.60\), the incorrect calculation \(0.50(0.15)+0.40(0.60)=0.315\) labels prevalence as incidence. The corrected incidence per original cohort is only \(0.50(0.15)=0.075\); \(0.40(0.60)=0.240\) represents existing disability continuing. The full prevalence is \(0.075+0.240=0.315\).
Another important qualification concerns the meaning of a transition probability. In an observation-period model, \(p_{AD}\) is the probability of being in \(D\) at the next observation given state \(A\) now. If the question instead counts every onset event that may occur between observations, including an onset followed by recovery before the next observation, endpoint transition probabilities alone may not identify that event count. Match the calculation to the event definition in the question.
Check Your Understanding
For each problem, identify the requested denominator and distinguish new transitions from disability already present.
- A cohort is all active at the start of a year. The annual probabilities are \(A\to D=0.15\), \(A\to A=0.85\), \(D\to D=0.80\), and \(D\to A=0.20\). Find first-year incidence and prevalence, then second-year incidence and prevalence per original cohort. (Answer: First year \(0.1500\) and \(0.1500\); second year \(0.1275\) and \(0.2475\).)
- At the start of a period, \(70\%\) of a cohort is active and \(30\%\) disabled. The transition probabilities are \(A\to D=0.10\), \(A\to A=0.90\), \(D\to D=0.60\), and \(D\to A=0.40\). Find incidence per original cohort, conditional incidence among active members, and end-of-period prevalence. (Answer: \(0.0700\), \(0.1000\), and \(0.2500\).)
- A cohort has starting probabilities \((0.40,0.50,0.10)\) in \(A,D,X\). The transition rows are \((0.70,0.25,0.05)\), \((0.20,0.70,0.10)\), and \((0,0,1)\). Find new active-to-disabled transitions per original cohort, disabled prevalence per original cohort, and disabled prevalence among those in \(A\) or \(D\) at the end. (Answer: \(0.1000\), \(0.4500\), and \(0.4500/0.8300=0.5422\).)
- A cohort is initially all active. The first-period matrix has rows \((0.80,0.20)\) and \((0.50,0.50)\); the second-period matrix has rows \((0.90,0.10)\) and \((0.25,0.75)\), with states ordered \(A,D\). Find second-period incidence per original cohort and end-of-second-period prevalence. (Answer: \(0.0800\) and \(0.2300\).)
- At the start of a period, \(60\%\) of a cohort is active and \(40\%\) disabled. If \(p_{AD}=0.12\) and \(p_{DD}=0.75\), find incidence per original cohort and prevalence at the end. Explain which term in the prevalence calculation is not new incidence. (Answer: Incidence \(0.60(0.12)=0.0720\); prevalence \(0.0720+0.40(0.75)=0.3720\); the \(0.3000\) continuing-disability term is not new incidence.)